/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 97 (a) An automobile with tires of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 97

(a) An automobile with tires of radius \(32 \mathrm{cm}\) accelerates from 0 to 45 mph in 9.1 s. Find the angular acceleration of the tires. (b) How does your answer to part (a) change if the radius of the tires is halved?

Short Answer

Expert verified
(a) Angular acceleration is 6.909 rad/s虏. (b) With halved radius, it becomes 13.819 rad/s虏.

Step by step solution

01

Convert Velocity Units

First, we need to convert the final velocity from miles per hour (mph) to meters per second (m/s). The given speed is 45 mph. Use the conversion factor: 1 mph = 0.44704 m/s.\[45 \text{ mph} \times 0.44704 \frac{\text{m/s}}{\text{mph}} = 20.1168 \text{ m/s}\]
02

Calculate Linear Acceleration

The automobile reaches the final speed of 20.1168 m/s in 9.1 seconds, starting from rest. To find the linear acceleration \(a\), use the formula:\[a = \frac{{v_f - v_i}}{{t}} = \frac{{20.1168 \text{ m/s} - 0 \text{ m/s}}}{9.1 \text{ s}} = 2.211 \text{ m/s}^2\]
03

Use the Relationship Between Linear and Angular Quantities

The linear acceleration \(a\) is related to the angular acceleration \(\alpha\) by the formula:\[a = \alpha \cdot r\]where \(r\) is the radius of the tire, 32 cm or 0.32 m. Rearrange the formula to solve for \(\alpha\):\[\alpha = \frac{a}{r} = \frac{2.211 \text{ m/s}^2}{0.32 \text{ m}} = 6.909 \text{ rad/s}^2\]
04

Calculate Angular Acceleration with Halved Radius

If the radius of the tires is halved, the new radius \(r'\) becomes 16 cm or 0.16 m. Use the same formula for angular acceleration with the new radius:\[\alpha' = \frac{a}{r'} = \frac{2.211 \text{ m/s}^2}{0.16 \text{ m}} = 13.819 \text{ rad/s}^2\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Acceleration
Linear acceleration is the rate of change of velocity over time. It's a key concept in physics problems involving moving objects. In our exercise, the car starts from rest and reaches a speed. This change in speed is what we quantify using linear acceleration.

To find linear acceleration (\[a\]), we use the formula:
  • \[a = \frac{{v_f - v_i}}{{t}}\]
where \(v_f\) indicates the final velocity, \(v_i\) the initial velocity, and \(t\) represents time.

Here, the car's initial velocity was zero, and the final velocity was converted from mph to m/s (from Step 1). After applying the formula, we computed the linear acceleration to be 2.211 m/s虏. This means the car's speed increased by 2.211 meters per second every second over the 9.1 seconds it's accelerating.
Tire Radius
The tire radius plays a significant role when transitioning from linear to angular measures. In rotational motion, the radius is a key factor in determining angular quantities, such as angular acceleration.

In the exercise, the tire radius is initially given as 32 cm. It is important to convert this to meters (0.32 m) for consistency with other units. The radius directly affects how we calculate angular acceleration from linear acceleration. The relationship is expressed as:
  • \[a = \alpha \cdot r\]
This means when we solve for angular acceleration (\(\alpha\)), the radius \(r\) is used as a divisor.

Later, halving the radius (to 16 cm or 0.16 m) demonstrates how a smaller tire physically changes the rotational dynamics, resulting in a higher angular acceleration.
Unit Conversion
Unit conversion is crucial in physics problems to ensure accuracy in computation and interpretation of results. The exercise involved a conversion from miles per hour (mph) to meters per second (m/s), since standard physics calculations are typically performed in SI units.

To convert 45 mph to m/s, we used the conversion factor 1 mph = 0.44704 m/s.

This means multiplying 45 by 0.44704, which gives us a result of 20.1168 m/s. Maintaining the correct units throughout calculations prevents errors and ensures consistency, especially when using related formulas.
Physics Problems
Solving physics problems often involves a series of logical steps that connect different concepts. For our problem, we needed to understand both linear and angular dynamics.

We started with a practical move: converting velocity units so that they match the formula requirements. Then, we identified linear acceleration, which set the stage for calculating angular acceleration using the relationship between them.

The next twist came with the tire radius change, demonstrating how alterations in parameters shift the dynamics: - Adjusting the tire radius highlighted the relationship between size and rotational speed. - It showed how doubling or halving the radius can influence the observed outcome, such as doubling the angular acceleration when the radius is halved.
Physics problems like these train us to think systematically, connecting small changes to larger conceptual outcomes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A diver runs horizontally off the end of a diving tower \(3.0 \mathrm{m}\) above the surface of the water with an initial speed of \(2.6 \mathrm{m} / \mathrm{s} .\) During her fall she rotates with an average angular speed of \(2.2 \mathrm{rad} / \mathrm{s}\). (a) How many revolutions has she made when she hits the water? (b) How does your answer to part (a) depend on the diver 's initial speed? Explain.

After you pick up a spare, your bowling ball rolls without slipping back toward the ball rack with a linear speed of \(2.85 \mathrm{m} / \mathrm{s}\) (Figure \(10-23\) ). To reach the rack, the ball rolls up a ramp that rises through a vertical distance of \(0.53 \mathrm{m}\). (a) What is the linear speed of the ball when it reaches the top of the ramp? (b) If the radius of the ball were increased, would the speed found in part (a) increase, decrease, or stay the same? Explain.

When astronauts return from prolonged space flights, they often suffer from bone loss, resulting in brittle bones that may take weeks for their bodies to rebuild. One solution may be to expose astronauts to periods of substantial "g forces" in a centrifuge carried aboard their spaceship. To test this approach, NASA conducted a study in which four people spent 22 hours each in a compartment attached to the end of a 28 -foot arm that rotated with an angular speed of \(10.0 \mathrm{rpm}\). (a) What centripetal acceleration did these volunteers experience? Express your answer in terms of \(g .\) (b) What was their linear speed?

Angular Acceleration of the Crab Nebula The pulsar in the Crab nebula (Problem 9) was created by a supernova explosion that was observed on Earth in A.D. 1054 . Its current period of rotation \((33.0 \mathrm{ms})\) is observed to be increasing by \(1.26 \times 10^{-5}\) seconds per year. (a) What is the angular acceleration of the pulsar in rad/s \({ }^{2} ?\) (b) Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? (c) Under the same assumption, what was the period of the pulsar when it was created?

The rotor in a centrifuge has an initial angular speed of \(430 \mathrm{rad} / \mathrm{s} .\) After \(8.2 \mathrm{s}\) of constant angular acceleration, its angular speed has increased to 550 rad/s. During this time, what were (a) the angular acceleration of the rotor and (b) the angle through which it turned?
See all solutions

Recommended explanations on Physics Textbooks

Engineering Physics

Read Explanation

Magnetism

Read Explanation

Thermodynamics

Read Explanation

Electricity and Magnetism

Read Explanation

Geometrical and Physical Optics

Read Explanation

Force

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.